A NONABNORMAL SUBGROUP CONTAINED ONLY IN SELF-NORMALIZING SUBGROUPS IN A FINITE-GROUP

If U is abnormal in G, then N-G(V) = V for every V greater than or equ al to U. The converse is known to be true for solvable groups, but in Finite Soluble Groups, Doerk and Hawkes indicate that its truth is not known for G finite and nonsolvable. In this note, we provide a counte rexample. We prove: Theorem. In the unitary group G = U-3(3) there exi sts a nonabnormal subgroup U isomorphic to S-4 such that U less than o r equal to V implies N-G(V) = V.